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Theorem frgrncvvdeqlem1 28087
Description: Lemma 1 for frgrncvvdeq 28097. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem1 (𝜑𝑋𝑁)

Proof of Theorem frgrncvvdeqlem1
StepHypRef Expression
1 frgrncvvdeq.xy . . . 4 (𝜑𝑌𝐷)
2 df-nel 3119 . . . . 5 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
3 frgrncvvdeq.nx . . . . . 6 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2907 . . . . 5 (𝑌𝐷𝑌 ∈ (𝐺 NeighbVtx 𝑋))
52, 4xchbinx 337 . . . 4 (𝑌𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
61, 5sylib 221 . . 3 (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7 nbgrsym 27156 . . 3 (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
86, 7sylnibr 332 . 2 (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9 frgrncvvdeq.ny . . . 4 𝑁 = (𝐺 NeighbVtx 𝑌)
10 neleq2 3124 . . . 4 (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
119, 10ax-mp 5 . . 3 (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12 df-nel 3119 . . 3 (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
1311, 12bitri 278 . 2 (𝑋𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
148, 13sylibr 237 1 (𝜑𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2115  wne 3014  wnel 3118  {cpr 4552  cmpt 5132  cfv 6343  crio 7106  (class class class)co 7149  Vtxcvtx 26792  Edgcedg 26843   NeighbVtx cnbgr 27125   FriendGraph cfrgr 28046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-nbgr 27126
This theorem is referenced by:  frgrncvvdeqlem7  28093  frgrncvvdeqlem8  28094  frgrncvvdeqlem9  28095
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